Circular motion about a central point or axis

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Euler Rotattion for j$\geq$ 1

I am trying to understand Euler Angles as given in Sakurai. Specifically, with the case j=1, where it is stated ($\hbar=1$) For j=1 ONLY, we can replace exp(-i$J_y\beta$)$\rightarrow 1-J_y^2(1-\...
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69 views

2 men support a uniform horizontal beam at its 2 ends .If one of them lets go ,the force exerted by the beam on the other man will?

A) remain unaffected B) increase C) decrease D) become unequal to the force exterted by him on the beam This was a question in one of my books for mechanics, they solved the question using ...
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33 views

Rotation of Spin-operator

I have to calculate the rotated Spin operator and been given the equation for the Rotation Matrix $R(\Theta,\mathbf{n})$ as well as its action on an arbitrary vector $\mathrm{a}$: $R(\Theta,\mathbf{...
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51 views

Explaining conservation of angular momentum with a disconnect

A probe is rotating in space and an instrument comes loose and disconnects. I need to explain why the angular velocity $\omega$ of the probe does not change. This equation describes how the angular ...
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977 views

Is all angular momentum quantized?

Angular momentum is definitely quantized in elementary particles and electrons in atoms. Molecules also have characteristic rotation spectra. Is it true that all angular momentum is quantized, ...
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63 views

If, for a body rolling on an incline, the friction coefficient isn't enough to allow pure rolling will it still roll?

More specifically that it won't be pure rolling (obviously) but would it still have some rotational motion along with its translational motion? (if yes how would we write their mechanical equations). ...
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38 views

Why do linear momentum and angular momentum have to be conserved in ground frame only?

According the Newton's second law of translational and rotational motion respectively if the net external force /torque acting on a body is zero then the linear/angluar momentum is a constant . So ...
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Astrophysics of spirals [duplicate]

What accounts for the structure of spiral galaxies? I would seem that, considering gravity is proportional to distance, outer objects could not be connected in any way to inner objects (ie. a spiral) ...
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34 views

When applying the equation of torque and equating it to $I\alpha$ which moment of inertia do we take?

I believe $T=I_{cm}\alpha$, where $I_{cm}$ is the moment of inertia about centre of mass and $\alpha$ is the angular acceleration. But do we take $I_{cm}$ even if the torque has been taken about a ...
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23 views

Equation for Turning Projectile

I'm trying to cook up an equation that will give me position for a projectile which operates in the Newtonian Domain. So a the fall equation is: $$\mathbf{p_t} = \frac{\mathbf gt^2}{2} + \mathbf{v_0}...
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97 views

How is this seemingly counterintuitive result in a rotation problem explained?

In the classic spool problem, where a spool of string of mass M is unrolled with a force F, like in the diagram, after working through the equations with Newton's laws of rotation and translation, we ...
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237 views

Understanding rotational motion on a rough surface

I am having trouble gaining an intuitive understanding of rotational motion for an object rolling on rough surface, and I have a few questions regarding it. Let's say that there's a cylinder that ...
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2answers
98 views

Rotation of a system of two masses [closed]

A uniform disc of mass M and radius R is suspended vertically by a pivot at its periphery. A particle of mass M is fixed to the rim and raised to the highest point above its center. The system is ...
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162 views

What is the relativistic mass of this spinning ball? [closed]

Relativistic Mass is: $$ m_r = \frac{m}{\sqrt{1 - v^2/c^2}} $$ So Einstein says that the faster an object moves, the more mass it gains (relativistic mass). So suppose you have a spherical ball ...
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138 views

Tension in a rod rotating about a fixed point

A rod of mass $m$ and length $l$ is rising about a fixed point in the ceiling with an angular velocity $\omega$ as shown in the figure. Now, on taking a small element on the rod, the net tension ...
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1answer
119 views

Is angular acceleration same about all points of a rotating ball?

Suppose a ball is rotating due to force $F$ applied at its top (on a rough ground).There is pure rolling. In one case we write the equation w.r.t COM i.e $F.R=I(\alpha_1)$ and $F-f=ma$ and $a=R\alpha$...
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2answers
116 views

Prove that a translation operator times a reflection operator is unitary and Hermitian [closed]

I am trying to prove some properties of the product of the (unitary) translation operator $\hat{T}(a)\psi(x) = \psi(x-a)$ and the (Hermitian) reflection operator $\hat{R} \psi(x) = \psi(-x)$. In ...
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27 views

How to spatially rotate the frame of reference Faraday Tensor? Special Relativity and Galilei Transforms

Hey I'm currently just starting out doing special relativity and I stumbled upon following problem: If I have the faraday antisymmetric tensor $F^{\alpha \beta}$ and I perform a spatial rotation of ...
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2k views

Distance in relativistic circular motion in invariant spacetime

I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a ...
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48 views

Confusion in Special Relativity: Rotating frame of reference

Suppose we are observing a rotating frame from an inertial frame, free from gravity, and try to measure the circumference of a circle drawn in the rotating frame. Since our measuring rod would be ...
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1answer
97 views

How does the tension force of a massless string affect the speed of a mass on its end?

i,j, and k are the unit vectors in the following question, and $\omega$ is the angular velocity. Lets say you have a mass m attached to a mass-less string of length $L$ tied to a peg with no friction....
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163 views

Can a quantity have two units?

We know that Force has unit of newton and torque has unit of newton meter. Then if you define the energy, which has same magnitude of work then, $W=Fx$ has unit of Joule ( $J$ ) (or $Nm$ ) while $W=\...
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90 views

How to conserve angular momentum?

If i want to conserve angular momentum from the point of contact of the solid sphere and the ground how should I do it? I know we can use $mvh = I\omega + MvR$ ,if the collision is elastic. But in ...
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Dynamics of counter-rotating flywheels

I've wondered about this for ages. If we create a pair of flywheels that rotate in the opposite direction with the same angular momentum, but are co-located and have the same mass and inertial moment ...
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98 views

Moment of Inertia of a Motion Simulator Frame

I have a rather specific question. I'm trying to build a motion simulator for flight and racing sims, similar to this: https://www.youtube.com/watch?v=JQAZB3EnI_w . The basic concept is a platform ...
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integrating small angular velocities

I know that for a constant angular velocity the following is true: $R=e^{W t} R_0$ where $W$ is an angular velocity tensor, $t$ is a time, and $R$ is a rotation matrix I believe the following is ...
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119 views

When does angular position, or angular displacement, not obey the rules of vector addition?

The only examples I've found talk about rotating an object about one axis and then about another axis. When you reverse the order, the object ends up in a different position. However, as those ...
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350 views

On application of angular momentum conservation laws

Consider a situation like this: a massless ring is kept fixed at rest on a horizontal plane. A massless thin string attached at its one end to a point on the circumference of the ring while its other ...
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2answers
1k views

Instantaneous Centre of Rotation

Let's say a body is undergoing both rotational and translational motion. I know that ICR of the body as a whole will be the point about which the body is doing pure rotation, so basically will be the ...
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3answers
139 views

Is centripetal acceleration mandatory for circular motion?

If we consider a case where a closed circular wall is present, and inside the boundary, just adjacent to the inner wall, two particles are placed touching the surface of inner boundary such that the ...
2
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1answer
3k views

Degrees of freedom in a diatomic molecule [duplicate]

We know that a monatomic compound can only have 3 degrees of freedom as we can consider it to be a point mass. However now that we consider a diatomic molecule, there are 3 degrees of freedom in ...
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72 views

Is there an absolute accelerated frame of reference?

I know from special relativity and from a little common sense that there is no absolute inertial frame of reference; that is, physics acts the same no matter what velocity you go at. However, that ...
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1answer
120 views

What is the force that stops a rotating wheel eventually? [duplicate]

Assuming a car is traveling on a road. At some point, the driver stops accelerating the car, so it travels some distance and then stops. Since the car's wheels are round and are rotating without ...
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3answers
328 views

Sign of torque when rolling an object down an incline

Suppose you have an object rolling down the incline at 30 degrees. Given the point of contact is instantaneously at rest, I decided to analyse torques at that point. Therefore, the only force ...
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1answer
234 views

A smaller sphere purely rolling down without slipping over another larger sphere [closed]

In this question we have to find out the angular velocity of the smaller sphere about its own axis at the instant it leaves the surface of the larger sphere and it is given that the smaller sphere is ...
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0answers
57 views

Solving for position of a SO(3) rotating object, given the integrable functions for components of angular velocity along the principle axes

Assuming that you have approximated or solved the Euler's Equations for components of angular velocity along its principal axes of inertia $x$, $y$ and $z$ - i.e. in the coordinate system that is ...
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2answers
98 views

A Confusion in Rotational Dynamics

I am trying to analyse the following situation using classical mechanical concepts. Consider a a straight rod $AB$ of mass $M$ and length $L$ placed on a frictionless horizontal surface. A force $F$ ...
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1answer
202 views

rotation behavior of spinning discs

If two discs of equal mass and volume are parallel to each other but not touching, share an axle but using magnetics do not touch it, are not under gravity, and begin to rotate in opposite directions ...
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1answer
150 views

Confusing concepts in proof of spherical addition theorem

In http://scipp.ucsc.edu/~haber/ph116C/SphericalHarmonics_12.pdf, section 4, pages 6..9 is a proof of the spherical harmonics addition theorem. Page 8 has eq.(25), an application of Laplace series: ...
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874 views

In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted?

I was reading about the equipartition theorem and I got the following quotations from my books: A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis ...
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1answer
67 views

Work done by tension when a particle is tied to a vertical cylinder using a thread and given a velocity perpendicular to the thread

For a particle attached to a rope moving along a circle that has the length of rope as radius, the tension provides centripetal force and work done by tension is zero since velocity of particle is ...
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3answers
157 views

Motion of a rigid body: Why is motion only a compostition of two forms: rotation and Translation?

If you are in a 2-D space the motion of any rigrid body is composed of 2 different things, namely rotation and translation. If you are in 3-D space the motion of any rigrid body is also composed of 2 ...
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2answers
442 views

Is the spin 1/2 rotation matrix taken to be counterclockwise?

The spin 1/2 rotation matrix around the $z$-axis I worked out to be $$ e^{i\theta S_z}=\begin{pmatrix} \exp\frac{i\theta}{2}&0\\ 0&\exp\frac{-i\theta}{2}\\ \end{pmatrix} $$ Is this taken to ...
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1answer
386 views

Why do Uranus and Venus rotate from East to west?

Why do only Uranus and Venus rotate from East to west, while others rotate in opposite direction? Is it because of the angle of their tiltness? Or due to the process of their formation?
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2answers
160 views

Does a body always rotate purely about its center of mass? [duplicate]

For example : A rod is held stationary in vertical position on a smooth horizontal ground and then released. Now the center of mass has velocity and acceleration and every point of the rod has some ...
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1answer
48 views

What are the limitations of using instantaneous axis of rotation? [closed]

I know when the body rotatates as well as translates IAR or ICR shouldn't be used but I am not able to understand why?
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98 views

Resultant center of rotation due to multiple moments

Say, there are multiple moments (M1, M2, M3) acting on a body (of irregular shape) at points P1, P2, P3 respectively. The body is free to rotate about any point. Now, which resultant center (the ...
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1answer
51 views

Will a gas rotate as fast as the spherical container it is contained within?

Let's say I have a sealed spherical glass container 30 cm in diameter which contains plain air. The glass container is rotated about its axis at 1 revolution per minute. My question is, would the gas ...
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2answers
635 views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
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3answers
210 views

Why does earth spin counter-clockwise?

If we observe the earth from the north pole, we can see that the earth is rotating counter clock wise direction. The earth spins due to angular momentum, but why only in counter clock wise. Why ...